Sept 9 2015 International Macroeconomics Harvard

Prévia do material em texto

Consumption Correlation and Low
Interest Puzzles, Aggregation, Gains
from Risk Sharing
Kenneth Rogoff
Fall 2015
Low interest puzzle is most crisply defined in a complete markets model
(1) Basic complete markets model (introducing notation)
(2) International consumption correlations puzzle
(3) Shiller Securities
(4) Equity Premium and Low Interest Rate Puzzle
(5) Cost of Capital market exclusion (first pass)
2
1 A Simple Two period, Two State Complete
Markets Model (to introduce notation)
two states, two periods, time separable utility
1 = (1) + (1)[2(1)] + (2)[2(2)]
1.1 Budget Constraints (with Arrow-Debreu securities)
3
Table 5.4 Measures of V m, the Securitized Value of a Claim to a
Country’s Entire Future GDP, 1992 (billions of U.S. dollars)
Country V m Std.rm/ Country V m Std.rm/
Argentina 2,460 9.86 Nigeria 2,019 10.06
Australia 4,340 3.88 Pakistan 2,894 2.45
Brazil 10,032 8.88 Philippines 1,602 3.68
Canada 7,663 4.22 South Africa 1,722 8.98
France 12,901 5.38 Spain 6,721 6.30
Germany (West) 16,796 4.47 Sweden 1,972 5.70
India 20,378 4.32 Switzerland 1,911 5.30
Italy 11,540 4.68 Thailand 4,007 3.99
Japan 31,762 8.41 Turkey 3,868 3.38
Kenya 418 4.34 United Kingdom 13,495 1.46
Mexico 9,583 5.33 United States 82,075 2.03
Netherlands 3,607 4.68 Venezuela 2,501 6.87
Source: Methodology is based on Shiller (1993, ch. 4). Underlying annual real GDP data are from
Penn World Table, version 5.6. Standard deviations are on annual return (income plus appreciation) of
a perpetual claim to GDP.
a 1990 value based on 1950–90 data.
Foundations
ofInternationalM
acroeconom
ics
(246)
C
hapter
5
O
bstfeld
&
R
ogoff
©
1996
M
assachusetts
Institute
of
Technology
First period constraint, where (1) is the cost of a bond that pays one unit in
period 2, state 1, and  is the risk free interest rate.
(1)
1 + 
2(1) +
(2)
1 + 
2(2) = 1 − 1
Period 2 budget constraints
2() = 2() +2()  = 1 2
Present value budget constraint
1 +
(1)2(1) + (2)2(2)
1 + 
= 1 +
(1)2(1) + (2)2(2)
1 + 
4
1.2 Optimal Behavior
Unconstrained max. problem (substitute out budget constraints)
1 = 
"
1 −
(1)
1 + 
2(1)−
(2)
1 + 
2(2)
#
+
2X
=1
()[2() +2()]
First order "Euler" conditions
()
1 + 
0(1) = ()0[2()]  = 1 2
Rearrange to show that MRS between 1 and 2 equals price
()0[2()]
0(1)
=
()
(1 + )
 = 1 2
5
Or, adding these two conditions, yields
0(1) = (1 + )E1{0(2)}
E1{0(2)}
0(1)
=
1
1 + 
6
2 Two-Country, Two Period Model, Many States
Global equilibrium requires supply = demand in period 1
w1 = 1 + 
∗
1 = 1 + 
∗
1 = 
w
1
and in all possible period 2 states
2() + 
∗
2() = 2() + 
∗
2 ()  = 1 2  
As before the Euler condition for an individual country is
0(1) = (1 + )()0[2()]()
7
or, with CRRA utility 1−(1− ),
2() =
"
()(1 + )
()
#1

1
Adding across the home and foreign Euler conditions and imposing global goods
market equilibrium in both periods yields
 w2 () =
"
()(1 + )
()
#1

 w1  = 1 2  
which implies a date 1 price for security  of
()
1 + 
= ()
"
 w2 ()
 w1
#−
 = 1 2  
8
or combining the above equation for any two states yields
()
(0)
=
"
 w2 ()
 w2 (
0)
#−
× ()
(0)
Making use of the above equations and, the equilibrium condition
X
=1
() = 1
one can straightforwardly solve for the real interest rates as
9
1 +  =
³
 w1
´−

P§
=1 ()
h
 w2 ()
i−
Note again that the riskless bond is redundant with complete A-D securities
2.1 Equilibrium Consumption Levels
Under certain conditions, the complete markets model implies that equilibrium
consumption growth is perfectly correlated across countries. With many states,
the Euler conditions reduce to
10
()0 [2()]
0(1)
=
()
(1 + )
=
()0
h
∗2()
i
0(∗1)
and
()0 [2()]
(0)0 [2(0)]
=
()
(0)
=
()0
h
∗2()
i
(0)0
h
∗2(
0)
i
With identical constant relative risk aversion utility in both countries1−(1−
), we can solve to get
11
2()
2(0)
=
∗2()
∗2(
0)
=
 w2 ()
 w2 (
0)
and
2()
1
=
∗2()
∗1
=
 w2 ()
 w1
which implies that home consumption is a constant fraction of world output
2()
 w2 ()
=
2(0)
 w2 (
0)
∗2()
 w2 ()
=
∗2(
0)
 w2 (
0)
12
2()
 w2 ()
=  =
1
 w1
∗2()
 w2 ()
= 1−  = 
∗
1
 w1
For countries with different relative risk aversion, and different subjective dis-
count factors, we can still get the generalization
log
"
2()
1
#
=
Ã


!
log
"
2 ()
1
#
+
1

log
Ã


!
which, unfortunately, fails miserably
13
Table 5.1 Consumption and Output: Correlations between
Domestic and World Growth Rates, 1973–92
Country Corr
(Oc, Ocw� Corr ( Oy, Oyw�
Canada 0.56 0.70
France 0.45 0.60
Germany 0.63 0.70
Italy 0.27 0.51
Japan 0.38 0.46
United Kingdom 0.63 0.62
United States 0.52 0.68
OECD average 0.43 0.52
Developing country average −0.10 0.05
Note: The numbers Corr.Oc, Ocw/ and Corr. Oy, Oyw/ are the simple correlation coefficients between the
annual change in the natural logarithm of a country’s real per capita consumption (or output) and
the annual change in the natural logarithm of the rest of the world’s real per capita consumption (or
output), with the “world” defined as the 35 benchmark countries in the PennWorld Table (version 5.6).
Average correlations are population-weighted averages of individual country correlations. The OECD
average excludes Mexico.
Foundations
ofInternationalM
acroeconom
ics
(243)
C
hapter
5
O
bstfeld
&
R
ogoff
©
1996
M
assachusetts
Institute
of
Technology
2.2 Rationale for the Representative-Agent Assumption
First assume agents have identical generalized CRRA utility but different wealth
levels
() =
(0 + 1)1−
1− 
Making use of our earlier bond Euler equations, we get
³
0 + 1

1
´−
=
⎡
⎣
(1 + )()
h
+ 12()
i
()
⎤
⎦
−
 = 1 2  
14
or
0 + 1

1 =
"
(1 + )()
()
#−1 h
0 + 1

2()
i
 = 1 2  
Sum over all agents, divide by  (number of agents) and raise result to the
power 1
(0 + 11)
− =
(1 + )() [0 + 12()]
()
−
Thus prices can behave as if there is a representative agent.
15
When agents do not have identical utility functions, we can still sometimes
aggregates using geometric averages instead of arithmetic averages. Define
˜ ≡ Π=1()
1

Assume agents have distinct CRRA utility function with different values of .
Then the individual Euler conditions for agent  is given by
2() =
"
()(1 + )
()
# 1

1
raising both sides to the 1 (where  is the number of individuals) and then
multiplying the resulting individual equations, one can obtain.
16
˜2() = Π

=1
"
()(1 + )
()
# 1

˜1
or
()
1 + 
= ()
"
˜2()
˜1
#−˜
where ˜ is the harmonic mean
˜ ≡ 11

P
=1
1

17
3 Consumption-Based Capital Asset Pricing Model
Extending the two period multi-state multi-country model, consider an asset
 with period 1 price   that pays  2 () in state of nature . The value of
the asset is the cost of purchasingArrow-Debreu bonds with the same payout
vector;
  =
§X
=1
(
()0[2()]
0(1)
)
 2 ()
where consumption can refer to any individual Because of complete markets,
it does not matter which individual we use to evaluate risky asset  . This
18
equation can alternatively be written as
  = 1
(
0[2()]
0(1)
 2
)
or equivalently
  = 1
(
0(2)
0(1)
)
1

2 + 1
(
0(2)
0(1)
)
  2
. Note that with constant relative risk aversion utility, we can replace individual
consumption with world consumption. Note that by the bond Euler equation,
1
(
0(2)
0(1)
)
=
1
1 + 
19
Then, defining the ex-post return on the above risky asset as
 =
 2 −  
 
we can arrive at the standard equation for the consumption based capital asset
pricing model
1(
)−  = −(1 + )1
(
0(2)
0(1)

)
  − 
20
Post-War Financial Repression?
Real US Stock, Bond Returns: 1802-1992
PERIOD Stocks Short Bonds Long Bonds
1802-1992 6.7 2.9 5.4
1871-1992 6.6 1.7 2.6
1802-1970 7.0 5.1 4.8
1871-1925 6.6 3.2 3.7
1926-1992 6.6 0.5 1.7
1946-1992 6.6 0.4 0.4
Source: Obstfeld and Rogoff, 1996, Siegel, 1995
4 The Equity Premium Puzzle
Mehra and Prescott (1987) first pointed out that the CCAPM fails very badly
to explain the excess return on stocks over bonds measured over long periods,
using 1889-1978 data.. Assume a constant relative risk aversion utility function
in the preceding consumption CAPM equation
E1{}−  = −(1 + )Cov1
⎧
⎨
⎩
Ã
2
1
!−
  − 
⎫
⎬
⎭
To do a simple empirical calculation, we approximate the function

Ã
2
1
 
!
≡ 
Ã
2
1
!−
( − E1)
21
in the neighborhood of 21 = 1 and  = E1. (Note: there are
approaches to this approximation though all yield the same general approach

Ã
2
1
 
!
≈ ( − E1)− 
Ã
2
1
− 1
!
( − E1)
Taking conditional (period 1) expectations of both sides yields
E1
Ã
2
1
 
!
= Cov1
⎧
⎨
⎩
Ã
2
1
!−
  − 
⎫
⎬
⎭
≈ −Cov1
(
2
1
− 1 ( − )
)
22
With CRRA utility, the CCAPM equation can be thus approximated as
E1 {}−  = (1 + )Cov1
(
2
1
− 1  − 
)
= (1 + )Std1
(
2
1
− 1
)
Std1 { − }
where  ≡ Corr1(21  − ) is the conditional correlation coefficient
between consumption growth and the excess return on equity, and Std1 is the
standard deviation of consumption growth. Mankiw and Zeldes 1991) find
that  = 04 in the Mehra Prescott data set, and the standard deviation of
consumption growth is 0036 per year, and the standard deviation of excess
23
returns on equity were 0167 per year. Assuming (1 + ) = 1, we find that
the model only fits the observed excess equity return of
E{}−  = 00698− 00080 = 00618
if  = 26, seemingly an implausible estimate
24
Habit persistence
() =
( −)1−
1− 
where 
 = (1− )−1 + −1 0     1
This approach helps explain equity premium since individuals are very averse to
big changes in consumption but then the MRS of consumption would be very
volatile and the risk free interest rate would be counterfactualy volatile.
25
Other explanations of equity premium puzzle:
"Keeping up with Joneses"
Transactions costs
Non-diversifiable consumption
Tail risk (Barro-Reitz)
Uncertainty about Future Trend Growth
26
5 The Low Interest Rate Puzzle
With CRRA utility, the stochastic Euler equation can be written
1 +  =
1

n
(+1 )
−
o
 ≈ log(1 + ) = 
(
log(
+1

)
)
− 
2
2
 
(
log(
+1

)
)
− log 
In Mehra Prescott sample, mean per capita consumption growth is 0.018 per
year with variance 0.0013. With  = 2, then even with  = 1, model predicts
that riskless rate is 3.34, but sample mean is only 0.8
27
THE DROP IN US REAL INTEREST RATES
Ben Bernanke Blog, March 30 2015
Growth Rates and Rates of Return for OECD 
Countries, 1870-2006 (or shorter samples)
Growth rates Real rates of return
Country ∆c/c ∆y/y ∆C/C ∆Y/Y stocks bills bonds
Australia 0.015 0.016 0.032 0.033 0.103 0.013 0.035
Canada 0.019 0.021 0.036 0.038 0.074 0.013 0.038
Denmark 0.016 0.019 0.024 0.027 0.071 0.032 0.039
France 0.016 0.019 0.020 0.023 0.060 -0.008 0.007
Germany 0.019 0.021 0.025 0.027 0.076 -0.015 -0.001*
Italy 0.017 0.021 0.023 0.027 0.053 0.005 0.017
Japan 0.025 0.028 0.035 0.038 0.093 0.004 0.031
Norway 0.019 0.023 0.027 0.031 0.072 0.021 0.028
Sweden 0.021 0.023 0.027 0.029 0.092 0.025 0.032*
U.K. 0.015 0.016 0.019 0.020 0.064 0.018 0.028
U.S. 0.019 0.022 0.033 0.036 0.083 0.020 0.027
Means 0.018 0.021 0.027 0.030 0.076 0.012 0.026
See Robert Barro QJE 2006
ROBERT BARRO QJE 2006 “rare disaster” model
log(Yt+1) = log(Yt) + g + ut+1 + vt+1
g is the deterministic part of growth process,
u is the i i d shock
Probability 1-p, vt+1=0
u is the i.i.d shock
Probability p, vt+1= log (1-b), 0<b<1
P , the probability of disaster is small, but b is large
• Barro’s 2006 QJE paper calibrates b and p byBarro s 2006 QJE paper calibrates b and p by 
looking at disasters where b > .15 using 
cumulative loss in output over the potentiallycumulative loss in output over the potentially 
multi-year catastrophe. (For further work, see 
Barro and Ursala (2008) and Barro NakamuraBarro and Ursala (2008) and Barro, Nakamura, 
Steinson and Ursala (2012).
ROBERT BARROS list of the Main Economic Crises ofROBERT BARROS list of the Main Economic Crises of 
20th Century
For real per capita consumer expenditure:
• WWII: 23 cases, average 34%
• WWI: 20 cases, average 24%
G t D i 18 21%• Great Depression: 18 cases, average 21%
• 1920s (influenza): 11 cases, average 18%
• Post-WWII: 38 cases average 18% (only 9 in• Post WWII: 38 cases, average 18% (only 9 in
tranquil OECD)
• Pre-1914: 21 cases, average 16%
Distribution of C Disasters
0
4
8
12
16
20
0.1 0.2 0.3 0.4 0.5 0.6 0.7
C-disaster size (N=95, mean=0.219)
See Robert Barro QJE 2006
Distribution of GDP Disasters
0
10
20
30
40
50
0.1 0.2 0.3 0.4 0.5 0.6 0.7
GDP-disaster size (N=152, mean=0.207)
See Robert Barro QJE 2006
Problems with basic formulation
• Power utility function has some counter-
factual predictions for rare disasters.
• Biggest problem is that rare disasters bid UP 
stock prices as people seek to save when the 
disaster starts
• Barro, Nakamura, Steinson and Ursala (2012) 
use Epstein Zinn (also Weil) formulation to 
separate risk aversion and intertemporal
substition.
Other refinements
• Barro looks at peak to trough fall in output. 
But actually output declines initially by more 
than in long run (global average is 30% in 
short run but 14% in long run
• Disaster risk is time varying
• Disaster risk is correlated across countries